Significance of negative eccentricity

It actually doesn't change anything; if you look at the equations for finding the eccentricity, it is the square root of something, and since there are two square roots of a number, it can be both negative or positive.

If you make a plot of a conic section based on the eccentricity, it doesn't change at all. Plot it on desmos graphing calculator using the formula r=$\frac{d}{1-dcos(\theta)}$ where d is the eccentricity (polar coordinates). Nothing happens if you negate the eccentricity.

formula for eccentricity